Functional calculus for the Ornstein-Uhlenbeck operator

Let γ be the Gauss measure on Rd and L the Ornstein-Uhlenbeck operator, which is self adjoint in L2(γ). For every p in (1, ∞), p≠2, set φ*p=arcsin2/p-1 and consider the sector Sφ*p={z∈C:argz<φ*p}. The main result of this paper is that if M is a bounded holomorphic function on Sφ*p whose boundary values on ∂Sφ*p satisfy suitable Hörmander type conditions, then the spectral operator M(L) extends to a bounded operator on Lp(γ) and hence on Lq(γ) for all q such that 1/q-1/2≤1/p-1/2. The result is sharp, in the sense that L does not admit a bounded holomorphic functional calculus in a sector smaller than Sφ*p. © 2001 Academic Press.